Optimal. Leaf size=154 \[ \frac {(6 A+8 B+13 C) \tan (c+d x)}{105 d \left (a^4 \sec (c+d x)+a^4\right )}+\frac {(6 A+8 B+13 C) \tan (c+d x)}{105 d \left (a^2 \sec (c+d x)+a^2\right )^2}+\frac {(8 A-B-6 C) \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac {(A-B+C) \tan (c+d x) \sec (c+d x)}{7 d (a \sec (c+d x)+a)^4} \]
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Rubi [A] time = 0.26, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {4078, 4000, 3796, 3794} \[ \frac {(6 A+8 B+13 C) \tan (c+d x)}{105 d \left (a^4 \sec (c+d x)+a^4\right )}+\frac {(6 A+8 B+13 C) \tan (c+d x)}{105 d \left (a^2 \sec (c+d x)+a^2\right )^2}+\frac {(8 A-B-6 C) \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac {(A-B+C) \tan (c+d x) \sec (c+d x)}{7 d (a \sec (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 3794
Rule 3796
Rule 4000
Rule 4078
Rubi steps
\begin {align*} \int \frac {\sec (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx &=-\frac {(A-B+C) \sec (c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {\int \frac {\sec (c+d x) (a (6 A+B-C)-a (2 A-2 B-5 C) \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac {(A-B+C) \sec (c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(8 A-B-6 C) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {(6 A+8 B+13 C) \int \frac {\sec (c+d x)}{(a+a \sec (c+d x))^2} \, dx}{35 a^2}\\ &=-\frac {(A-B+C) \sec (c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(8 A-B-6 C) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {(6 A+8 B+13 C) \tan (c+d x)}{105 d \left (a^2+a^2 \sec (c+d x)\right )^2}+\frac {(6 A+8 B+13 C) \int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{105 a^3}\\ &=-\frac {(A-B+C) \sec (c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(8 A-B-6 C) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {(6 A+8 B+13 C) \tan (c+d x)}{105 d \left (a^2+a^2 \sec (c+d x)\right )^2}+\frac {(6 A+8 B+13 C) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.79, size = 231, normalized size = 1.50 \[ \frac {\sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right ) \left (-35 (18 A+5 B+4 C) \sin \left (c+\frac {d x}{2}\right )+70 (9 A+4 B+2 C) \sin \left (\frac {d x}{2}\right )+441 A \sin \left (c+\frac {3 d x}{2}\right )-315 A \sin \left (2 c+\frac {3 d x}{2}\right )+147 A \sin \left (2 c+\frac {5 d x}{2}\right )-105 A \sin \left (3 c+\frac {5 d x}{2}\right )+36 A \sin \left (3 c+\frac {7 d x}{2}\right )+168 B \sin \left (c+\frac {3 d x}{2}\right )-105 B \sin \left (2 c+\frac {3 d x}{2}\right )+91 B \sin \left (2 c+\frac {5 d x}{2}\right )+13 B \sin \left (3 c+\frac {7 d x}{2}\right )+168 C \sin \left (c+\frac {3 d x}{2}\right )+56 C \sin \left (2 c+\frac {5 d x}{2}\right )+8 C \sin \left (3 c+\frac {7 d x}{2}\right )\right )}{6720 a^4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 135, normalized size = 0.88 \[ \frac {{\left ({\left (36 \, A + 13 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (39 \, A + 52 \, B + 32 \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (6 \, A + 8 \, B + 13 \, C\right )} \cos \left (d x + c\right ) + 6 \, A + 8 \, B + 13 \, C\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 171, normalized size = 1.11 \[ -\frac {15 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 63 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 21 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 21 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 35 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 35 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 105 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 105 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{840 \, a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.81, size = 108, normalized size = 0.70 \[ \frac {\frac {\left (-A +B -C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {\left (-C +3 A -B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\left (-3 A -B +C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 259, normalized size = 1.68 \[ \frac {\frac {C {\left (\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}} + \frac {B {\left (\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}} + \frac {3 \, A {\left (\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.31, size = 99, normalized size = 0.64 \[ \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A+B+C\right )}{8\,a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B+C\right )}{56\,a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (B-3\,A+C\right )}{40\,a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (3\,A+B-C\right )}{24\,a^4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {A \sec {\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \sec ^{2}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \sec ^{3}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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